7 PRIMITIVES AND INTEGRALS 169

2. (a) Let / be a regulated function defined in a compact interval I = [a, b]. Show
that for any s > 0, there exists a continuous function g defined in I and such that

j» 2}

(b) Suppose/takes its values in E; let h be a regulated function defined in I and taking
its values in F, and let (x, y) -+ [x • y] be a continuous bilinear mapping of E x F into
G (E, F, G Banach spaces). Show that

lim f [f(t) - h(t + s)] dt == f\f(t) - h(t)] dt.

s-cO, s>0 Ja Jfl

(c) Show that

lim f /(/) sin nt dt = lim f /(/) cos nt dt = 0, lim f /(/) |sin nt\ dt=- f /(/) dt.

II-K» Ja n-+oo Ja n-»co Ja TTja

(d) Let u be a primitive of/, and suppose u(I) is contained in a ball B <= E. Show that if
g is a monotone function in I, then there exists a number c e B such that

f /('MO dt = (u(b) - c)g(b) + (c- u(a

Ja

In particular, if /is a real regulated function, there exists s e I such that

("the second mean value theorem").

(For all these properties, use the same method as in Problem 1 .)

3. Let /be a regulated function defined in a compact interval I = [a, b]. For any integer
n > 0 and any integer k such that 0 ^ k < «, let xk = a + k((b — a)/n); let

(a) Suppose /has a continuous derivative in I. Show that

(Write r(ri) = y (/fe+i) ~/(0) <#J use tne mean value theorem and Problem 1.)

fc=oJxft

(b) Suppose /is an increasing real function in I; show that

(c) Give an example of an increasing continuous function /in I such that nr(ri) does
not tend to ((b — a)/2)(f(b) — f(a)) when n tends to +00. (Take for / the limit of
a sequence (/„) of increasing continuous piecewise linear functions, satisfying the con-
ditions

- T f /») dt*i(b- a)(fn(b) -fn(a))

J«

2"

and

for